Mode interference effect in coherent electron focusing

EUROPHYSICS LETTERS 15 October 1988 Europhys. Lett., 7 (4), pp. 359-364 (1988)

Mode Interference Effect in Coherent Electron Focusing.

C. W. J. BEENAKKER(*), H. VAN HOUTEN(*)(

) and B. J. VAN WEES(**)

(received 9 May 1988; accepted in final form 28 July 1988) PACS. 72.20M - Galvanomagnetic and other magnetotransport effects. PACS. 73.20 - Electronic surface states.

PACS. 73.40L - Semiconductor-to-semiconductor contacts, p-n junctions, and heterojunctions.

Abstract. - A novel quantum interference effect in ballistic transport is described: the

interference of coherently excited magnetic edge states in a two-dimensional electron gas. The effect explains the characteristic features of the unexpected fine structure observed recently in an electron focusing experiment.

Advances in semiconductor technology have brought within reach the realization of electron optics in the solid state. In a two-dimensional electron gas (2DEG) narrow channels have been defined lithographically, through which electrons propagate with minimal scattering äs in a wave guide [l, 2]. Short constrictions with a variable width of the order of the Fermi wavelength (quantum point contacts) show a quantized conductance of 2ezlh per occupied subband or wave guide mode [3,4]. The first experimental observation in a 2DEG of electron focusing by a magnetic field was reported in ref. [5], with quantum point contacts äs injector and collector of ballistic electrons. At low temperatures fine structure was seen in the focusing peaks (not expected from earlier experiments in metals), which suggested that coherent electron focusing had been realized. This intriguing possibility is investigated theoretically in this letter. We shall demonstrate that the characteristic features of the focusing spectrum can be understood äs an interference of coherently excited magnetic edge states.

The technique of electron focusing, pioneered in metals by Sharvin[6] and Tsoi[7], has become a powerful tool to investigate Fermi surfaces, boundary scattering, and the electron-phonon interaction [8]. Tsoi's transverse geometry (fig. 1) consists of two point contacts (injector and collector) on the same boundary in a perpendicular magnetic field. Electrons at the Fermi energy EF are injected ballistically through the injector, and move in a skipping orbit along the boundary towards the collector, which serves äs a voltage probe (drawing no net current). Classically, peaks in the collector voltage vs. magnetic-field curve

360 EUROPHYSICS LEITERS

Fig. 1. - Top: Skipping orbits in a 2DEG. The gate defming the injector (i) and collector (c) point contaets and the 2DEG boundary is shown schematically in black. For clarity the trajectories are drawn only up to the third specular reflection. Bottom: calculated location of the caustic curves. occur when a focus of the trajectories shifts past the collector. In the 2DEG of a GaAs-AlGaAs heterostructure the Fermi surface is simply a circle. The corresponding skipping orbits are shown in fig. 1. The collector coincides with a focal point when its Separation L from the injector is an integer multiple of twice the cyclotron radius Zcyd = hkvleB. Focusing thus takes place at magnetic fields B which are multiples of

cus = 2hkF/eL, (D

with kF the Fermi wave vector. A simple calculation [9] of the fraction of trajectories which reach the collector from the injector predicts a series of equidistant peaks of equal height, above a monotonously increasing baseline. The p-th peak is due to electrons which have made p — l specular reflections at the boundary. Such a classical focusing spectrum is commonly observed in metals, albeit with a decreasing height of subsequent peaks because of partially diffuse scattering [7-10].

In the 2DEG a strikingly different focusing spectrum is found [5]. At high temperatures (4 K) a series of focusing peaks is indeed observed at multiples of ßfocus = 0.066T,

demonstrating the ballistic injection of electrons with specular reflection at the 2DEG boundary. However, upon lowering the temperature (down to 30 mK) a fine structure develops on the low-field focusing peaks, which is most pronounced äs the width of the point contaets is reduced to about a Fermi wavelength (AF Ξ 2nlkv ~ 40 nm). The fine structure

C. W. J. BEENAKKER et dl.: MODE INTERFERENCE EFFECT ETC. 361

0 50 frequency O / T )

Fig. 2. - Magnetic-field dependence of the collector voltage Vc (divided by the injected current /,),

measured with the double point contact device of ref. [5] at T = 50 mK. Shown are the results for two gate voltages Vg=- 1.53 V (lower trace), and Vg = - 1.22 V (upper trace). The omitted field region below 0.05 T contains a magnetoresistance originating in the ohmic contacts, see ref. [5]. The inset gives the Fourier transform power spectrum of Vc for B>0.4T (dashed curve: Vg= — 1.53 V, solid curve: Vg=- 1.22V).

the 2DEG boundary; see fig. 2, where experimental results for two different gate voltages are shown. Note that increasing the negative gate voltage has the effect of reducing the width of both point contacts. A Fourier transform of the spectra for B > 0.4 T (inset in fig. 2) shows that the dominant periodicity (0.06 ± 0.01) T of the high-field oscillations is approximately the same äs the periodicity 5focus of the low-field focusing peaks. However,

the amplitude is much larger.

To explain these observations it is necessary to go beyond the classical description. We first present a simple qualitative argument. Quantum ballistic transport along the 2DEG boundary takes place via magnetic edge states [11,12], which are the propagating modes of this problem. The modes at the Fermi level are labelled by a quantum number n = l, 2,..., nmax. Since the injector has a width below λρ, it excites these modes coherently.

For kFL » l the interference of modes at the collector is dominated by their rapidly varying

362 EUROPHYSICS LETTERS potential). Simple geometry shows that this requires that(:)

2π L- n n=l,2...nm„, (2)

with nmax the largest integer smaller than (1/2) /CF icyci + 1/4. The dependence on n of the

phase knL is close to linear in a broad interval. This follows from expansion of eq. (2) around

a„ = 0, which gives

ι

knL = constant - 27m-— + kF L x order -^ - . (3)

•Dfocus \ max

If B/Biocus is an integer, a fraction of order (l//cFL)1/3 of the wmax edge states interfere

constructively at the collector. (The edge states outside the domain of linear %-dependence of the phase give rise to additional interference structure which, however, does not have a simple periodicity.) Because of the 1/3 power, this is a substantial fraction even for the large /cFL~450 of the experiment. The resulting mode interference oscillations with 5focus

-periodicity can become much larger than the classical focusing peaks. To demonstrate this, we now calculate in WKB approximation the wave function Ψ in the 2DEG.

We consider a point-dipole injector(2) and determine \dW/dx2 at the coordinates (x, y) = (0, L) of the collector — but unperturbed by its presence. We do not attempt to

actually calculate the transmission probability from injector to collector, since this quantity is sensitive to the detailed form of the gate potential defining the point contacts and the 2DEG boundary — which we do not know. (Such a calculation would also have to take into account the reduced carrier density in the point contact region.) For point contacts with a width of the order of λρ, the ß-dependence of the collector voltage is determined in first

approximation by the unperturbed probability density at an infinitesimal distance from the collector. Since for an infinite barrier potential both ψ and dY/dy vanish at x = 0, this density is proportional to our calculated |3¥73ic|2. In the WKB approximation [14], the wave

function W(x, y) is the sum over all classical trajectories from injector to the point (x, y) of an amplitude factor times a phase factor exp [ίφ]. The amplitude factor is inversely proportional to the square root of the cross-section of a particle flux tube containing the trajectory, äs required by current conservation. The phase increment φ acquired along the trajectory is the sum of four terms: 1) a path length term kF l, with l the length of the trajectory. 2) The

Aharonov-Bohm phase (- e/h)$dl · A, given by the integral of the vector potential along the trajectory. In the gauge A = (0, Bx, 0) this term equals — eBO/h, with 0 the area between the trajectory and the boundary at x = 0. 3) A phase shift of π for each specular reflection at the boundary. 4) A phase shift of — π/2 for each passage through a caustic, which is a point at which the cross-section of the flux tube is reduced to zero (see fig. 1). In view of the long mean free path l,^ ~ 9 μ-m in the experiment [5], we do not include the effects of impurity scattering in our calculation. We have found that taking into account impurity scattering in C1) We neglect spin-splitting, since the Zeeman energy gpBB^10~3EF in the field ränge

considered. We also neglect a possible ß-dependence of kv. In the bulk of the 2DEG, pinning of EF at

Landau levels leads to a modulation of kF by up to 10% at l T. Near the boundary, however, we expect

this effect to be much reduced because edge states fill the energy gap between Landau levels. Note also that since the fcF-modulation is periodic in 1/B, it does not lead to a defmite .B-periodicity of the

collector voltage.

(2) Bipolar injection CFoceosa) was chosen instead of isotropic injection, to satisfy the boundary

C. W. J. BEENAKKER et al.: MODE INTERFERENCE EFFECT ETC. 363

an averaged way, by weighing the contribution of the trajectories to Ψ with a factor exp [— 1/21^], does not significantly affect our results.

The resulting magnetic-field dependence of |3¥Ύ3χ|2 is shown in fig. 3 (bottom), for the

experimental values L = 3.0 μηι and kF = 1.5 · 108ηι-1 of ref. [5]. The most rapid oscillations

were eliminated by averaging L over an interval of 100 nm, which corresponds roughly to

Fig. 3. - The lower curve shows the magnetic-field dependence of the unperturbed probability density at an infinitesimal distance from the collector, determined by |3¥73x|2. The top curve results if only

the incoherent contributions are retained (no interference of skipping orbits).

the combined width of the point contacts. Also plotted in fig. 3 (top) is the incoherent contribution to |3Ϋ79«2, without the interference of different trajectories, which shows

simply the peaks from classical electron focusing at multiples of 5focus. Interference effects

give rise to fine structure on the focusing peaks at low magnetic fields, which grows in amplitude with increasing field. It is apparent from fig. 3 (and confirmed by Fourier analysis) that the large-amplitude high-field oscülations have the same periodicity äs the

smaller low-field focusing peaks—äs observed experimentally, and consistent with the mode interference argument given above. This is the main result of our calculation, which we have found to be insensitive to details of the point contact modelling. (Insensitive, for example, to assuming isotropic instead of dipolar injection.) The relation between the above description in terms of interfering skipping orbits and the description in terms of interfering edge states used in the qualitative argument can be made explicit, by transforming the sum over trajectories into a sum over modes using the method of stationary phase. We have verified in this way that the phases of the modes are indeed determined by eq. (2). These two alternative representations of the quantum-mechanical transport problem are the analogues of the classical ray and mode descriptions of propagation in a wave guide. In this context the edge states correspond to Lord Rayleigh's «whispering gallery» waves [15].

364 EUROPHYSICS LETTERS orbit. We do not see how this can be reconciled with the fact that the probability density

\Ψηζ of an individual edge state is i/-independent (since Yn(x,y)=fn(x) exp [ik„ y]).

A quantitative comparison between theory and experiment requires a more detailed modelling of the point contacts and gate potential. The appearance of high-field oscillations with the focusing periodicity but with much larger amplitude is, however, characteristic of the mode interference mechanism proposed in this letter (and is indeed the feature which experimentally is insensitive to small changes in gate voltage). This novel quantum interference effect in ballistic transport described here for the double point contact geometry of ref. [5] may also play a role in the multi-probe «electron wave guides» [1] of current interest. Voltage fluctuations with a well-defined periodicity were found in such a device by Chang et al. [1], albeit in the regime where the transport was not fully ballistic. Our demonstration of coherent electron focusing shows that interference experiments can be realized using quantum point contacts äs monochromatic electron sources. These may be seen äs the first proven building blocks of electron optics in the solid state.

* * *

We have benefitted from frequent discussions on this subject with L. F. FEINER and M. F. H. SCHUURMANS.

REFERENCES

[1] TIMP G., CHANG A. M., MANKIEWICH P., BEHRINGER R., CUNNINGHAM J. E., CHANG T. Y. and HOWARD R. E., Phys. Rev. Lett., 59 (1987) 732; CHANG A. M. et al, Surf. Sei., 196 (1988) 46; TIMP G. et al., Phys. Rev. Lett., 60 (1988) 2081.

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[6] SHARVIN Yu. V., Z. Eksp. Teor. Fiz., 48 (1965) 984 (Sov. Phys. JETP, 21 (1965) 655). [7] Tsoi V. S., Pis'ma Z. Eksp. Teor. Fiz., 19 (1974) 114 (JETP Lett., 19 (1974) 70). [8] VAN SON P. C., VAN KEMPEN H. and WYDER P., Phys. Rev. Lett., 58 (1987) 1567. [9] BENISTANT P. A. M., Thesis, University of Nijmegen, The Netherlands (1984).

[10] Interesting deviations from the classical focusing spectrum (not related to the present work) can occur upon diffraction at a metal boundary, see BOZHKO S. L, SVEKLO I. F. and Tsoi V. S., Pis'ma Z. Eksp. Teor. Fiz., 40 (1984) 480 (JETP Lett., 40 (1985) 1313); HOEVERS H. F. C., private communication.

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[13] KOSEVICH A. M. and LIFSHITZ I. M., Z. Eksp. Teor. Fiz., 29 (1955) 743 (Sov. Phys. JETP, 2 (1956) 646).

[14] SCHULMAN L. S., Techniques and Applications of Path Integration (Wiley, New York, N. Y.) 1981.